Calculus/Calculus

QuestionA landscape architect plans to enclose a 4000 square foot rectangular region in a garden. She will use shrubs that cost $30 per foot along three sides and plants that cost $10 per foot along the fourth side. Determine a function giving the total cost of the project and then find the minimum cost.

AnswerLet x and y denote the lengths of the sides of the rectangle.

Then xy=400 =====> y=4000/x -------(1)

Total cost involved in enclosing rectangular region

C= 30(2x)+ 30(y) + 10 (y)

= 60x + 40y -----------(2)

Substituting (1) into (2) gives

C = 60x + 40( 4000/x) = 60x + 160000/x ---------(3)

When cost is minimized, dC/dx =0

Hence, 60 - 160000/(x^2) =0

160000/(x^2) = 60

x^2= 2666.667

x= 51.64 ft

Hence, minimum cost is given by substituting x= 51.64 ft into (3)

ie C =60(51.64) +160000/(51.64)

=$ 6196.77 (shown)

Note: I did not specifically assign the variables x and y to be the length and breadth of the rectangle. There is no need to.

An alternate cost function would be C = 30(2y) + 30x +10x = 60y +40x, however applying calculus to this expression once again would still yield the same end result.

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